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A weak space-time formulation for the linear stochastic heat equation

Journal article
Authors Stig Larsson
Matteo Molteni
Published in International Journal of Applied and Computational Mathematics
Volume 3
Issue 2
Pages 787-806
ISSN 2349-5103
Publication year 2017
Published at Department of Mathematical Sciences
Pages 787-806
Language en
Keywords Inf–sup theory, Stochastic linear heat equation, Additive noise, Linear multiplicative noise
Subject categories Probability Theory and Statistics, Applied mathematics, Mathematics


We apply the well-known Banach–Nečas–Babuška inf–sup theory in a stochastic setting to introduce a weak space-time formulation of the linear stochastic heat equation with additive noise. We give sufficient conditions on the data and on the covariance operator associated to the driving Wiener process, in order to have existence and uniqueness of the solution. We show the relation of the obtained solution to the mild solution and to the variational solution of the same problem. The spatial regularity of the solution is also discussed. Finally, an extension to the case of linear multiplicative noise is presented.

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